Higher monoidal injections and diagrammatic E n structures

Commutative rectification ofE_∞structures andE2structures was achieved in [SS12] and [SS16] respectively, by using diagram spaces and diagram categories with these struc-tures. The main objective of this paper is to continue this line of work and considerE_n structures on diagram categories and diagram spaces. Specifically we will use diagrams with ann-fold monoidal category, see Definition 5 in Preliminaries, as indexing category.

Proposition 4.2 in the paper shows that if we have a closed symmetric monoidal cate-gory C, where the underlying category is cocomplete, and we have a small strict n-fold monoidal category A, then the diagram category C^A inherits ann-fold monoidal struc-ture. Throughout this section ⊗ stands for the monoidal product in such a category C.

A crucial step in setting up the theory in this paper is to determine what we mean by anE_nstructure on a diagramX∈ C^A, whereAisn-fold monoidal. For this we use the notion of n-fold monoidal operads developed in [Sol]. In Section 4.16 we consider the n-fold monoidal operadE with underlying functor

E:Fn^op→Cat, a7→(a↓ Fn).

This is the operad that was denoted En in the previous paper. Recall that it is an n-fold version of the categorical Barratt-Eccles operad and that its algebras generalize the concept of strict n-fold monoidal categories, see Propositions 8.5 and 8.2 in [Sol]. The latter fact is also reflected in the symmetrization ofE, which we will discuss later in this presentation.

The action of ann-fold monoidal operadC, internal toC, on a diagramX∈ C^A, can be described in more explicit terms than the action in a general n-fold monoidal category.

According to Definition 4.7 in the paper it consists of a family of natural transformations θ_c:C(c)⊗X(a1)⊗ · · · ⊗X(ak)→X(^c(a1, . . . , a_k))

indexed by objects c ∈ Fn(k) and a_i ∈ Afor i = 1, . . . k. The object ^c(a1, . . . , a_k) is constructed as an n-fold monoidal product of the objectsa₁, . . . , a_k according to the structure ofc. The natural transformations must be unital, associative and equivariant as specified in the definition. A diagramXequipped with aC-action is called aC-algebra.

In order to relate this concept of Enstructures as algebras over then-fold monoidal op-eradEto traditionalE_nstructures we define symmetrization ofn-fold monoidal operads

in Section 4.8. An n-fold monoidal operad C internal to C has an underlying functor C: Fn^op → C. Left Kan extension along the canonical functor ς_n: Fn^op → Σ^op gives a functorςn!(C) : Σ^op → C natural in C. The n-fold monoidal operad structure on C induce a symmetric operad structure onς_n!(C), and this is what we call the symmetriza-tion ofC. Proposition 4.9 shows thatς_n!is the left adjoint in an adjunction between the category ofn-fold operads internal toC and symmetric operads internal toC:

ςn!: Opn(C)OpΣ(C) :ς_n^∗

As a corollary we get that, in a symmetric setting, C-algebras and ς_n!C-algebras are naturally isomorphic. In the result below the permutative category A is considered to have the canonical n-fold monoidal structure associated to a symmetric monoidal category, see Remark 7 in the Preliminaries.

Corollary (Corollary 4.11). LetA be a small permutative category and let C be an n-fold monoidal operad internal toC. Then the categories of algebrasC-C^Aandς_n!C-C^A are naturally isomorphic.

These results provide the justification for considering aC-algebra structure as an E_n structure, if the nerve of the symmetrization ofC is an E_n operad. As mentioned in Section 1.1.3 of the Preliminaries, the nerve of the symmetric operad Mn is an En

operad. Therefore the following result lets us consider the algebras over the n-fold monoidal operadE asE_n structures.

Proposition(Proposition 4.17). The symmetrization of then-fold monoidal operadE is isomorphic to the operadMngoverning n-fold monoidal categories.

For technical reasons we need to shift to using then-fold monoidal operadE^op. This is defined similarly toE, but with the opposite category at each level. The symmetriza-tion ofE^opisM^opn which is isomorphic toMn, soE^op-algebras are alsoEn structures.

Proposition 5.3 shows that for anE^op-algebraX inCat^A, the Grothendieck construc-tion AR

X inherits the structure of an E^op-algebra inCat, which is equivalent to the structure of ann-fold monoidal category by the above corollary.

We writeE^op-Cat^A for the category ofE^op-algebras inCat^A, and similarlyE^op-Cat for the category ofE^op-algebras inCat. A morphism inE^op-Catis a weak equivalence if the nerve of the underlying functor is a weak equivalence of simplicial sets. A morphism in E^op-Cat^Ais a weak equivalence if the induced functor on the Grothendieck construction is a weak equivalence. Localizing with respect to these classes of weak equivalences respectively, yields homotopy categories ofE^op-algebras. The main result of the paper

is that there is an equivalence between the homotopy categories ofE_nstructures inCat andCat^A.

Theorem (Theorem 5.11).LetAbe a small and strictn-fold monoidal category with contractible classifying space. Then the functors AR

and ∆ induce an equivalence be-tween the localized categories

AR

:E^op-Cat^A[w⁻¹_A ]'E^op-Cat[w⁻¹] : ∆.

So far the results mentioned have been about diagrams indexed by any small strict n-fold monoidal categoryA. Now we consider a specific indexing category we callIn, the category ofn-fold monoidal injections, see Definition 3.4 in the paper. The objects ofIn

are the objects of the freen-fold monoidal category on one element,Fn. A morphism in In, called ann-fold monoidal injection, consists of a pair of one morphism inFnand an injective order preserving function of ordered finite sets. This is similar to how a braided injection in the category B or an injection in the category I can be decomposed: A braided injection can be written as a pair of an element in the braid group and an order preserving function. An injection can be written as a pair of an element in the symmetric group and an order preserving function. In all three cases the number of elements in the domain of the order preserving function should match the permutation/braid/morphism in Fn. A further common trait of these categories is that they have similar universal properties. As per Remark 3.11 in this paper,Iis a free permutative category generated by the morphism 0→1 andBis a free braided strict monoidal category the morphism 0→1. ForInthe universal property is stated in the following result.

Proposition (Proposition 3.9). The categoryIn is the free n-fold monoidal category generated by the morphism 0→1.

A goal for future research is to study n-fold commutative monoids in Cat^In further.

Particularly to check if the homotopy category of n-fold commutative monoids inCat^In is equivalent to the homotopy category ofn-fold monoidal categories, localizing each of the categories with respect to the relevant weak equivalences. In this paper we have taken a step in that direction by showing that it is possible to realize a free n-fold monoidal category as the Grothendieck construction of an n-fold commutative monoid inCat^In: Given a categoryX with a distinguished object∗ ∈X, there is a functorX^•:I →Cat that maps p to the product category X^p and takes a morphism f:p → q in I to a functor f_∗:X^p → X^q. For a p-tuple of objects x = (x1, . . . , x_p), the components of f_∗x = y are given by y_j = x_i if f(i) = j and y_j = ∗ if j is not in the image of f.

Precomposing with the canonical functorςn: In→ I(see Corollary 3.10 in this paper), we get an element inCat^In. We can giveX^• ann-fold commutative monoid structure

where the product is induced by concatenation of tuples of objects, see the discussion at the start of Section 6 of this paper. LetX be a small category andX₊ the disjoint union ofXwith the terminal category. Treating the disjoint object as the distinguished object we form the Mn-algebraIn

R(X+)^•. The free n-fold monoidal category onX is Mn(X), whereMnis the monad associated to the symmetric operadMn. The inclusion X→ In

R(X+)^•induces a map ofMn-algebrasMn(X)→ In

R(X+)^•. Theorem(Theorem 6.6).The canonical map ofMn-algebras

Mn(X)→ In

R(X+)^•

is a weak equivalence.

The free n-fold monoidal category Mn(X) is therefore weakly equivalent to the Grothendieck construction of then-fold commutative monoid (X+)^•and we have a con-crete model forMn(X) inCat^In.

Analogously, in the simplicial set setting, we have the following result.

Theorem(Theorem 6.3). For a based simplicial setX there is a natural weak equiva-lenceρ:X_hI^•_n−→^∼ NMn∗(X) ofNMn-algebras.

2.2 Future research

Here we list some topics of interest for future research.

• It would be good to have more explicit examples ofn-fold monoidal operads. One idea is to try and make an n-fold monoidal operad version of the little n-cubes operad.

• Recall that an E_∞ operad is a Σ-free operad which is contractible at each level.

Fornequals 1 and 2,E1- andE2-operads can be modeled byA_∞- andB_∞-operads respectively. AnA_∞-operad is a non-Σ operad that is contractible at each level. A B_∞-operad is a braided operad such that each level is contractible, and the actions of the braid group at each level is free.

Ann-fold monoidalE_n-operad can be defined as ann-fold monoidal operad with a contractible space at each level, such that the underlying functor is cofibrant in a suitable model structure. The cofibrant condition is analogous to the condition that the group action should be free forE_∞andB_∞operads. It would be interest-ing to further examine the relationship betweenn-fold monoidalE_n-operads and symmetric monoidalE_n-operads.

• In [BM03] Berger and Moerdijk define model structures on operads internal to symmetric monoidal model categories, given that certain conditions are satisfied.

It would be interesting to see if a similar approach can be taken to define model structures onn-fold monoidal operads.

• In [Bat10] Batanin defines locally constant n-operads as higher braided operads.

Both n-fold monoidal operads and locally constant n-operads generalize non-Σ operads (n= 1), braided operads (n= 2) and symmetric operads (n =∞), but different aspects are generalized. It would be interesting to explore the relationship between these different generalizations.

• Finally, there is the continuation of the red thread that runs through this thesis, rectifyingE_n structures. In [SS] we introduced the categoryIn ofn-fold monoidal injections as ann-fold analog of the indexing categoriesI forI-spaces and Bfor B-spaces, used in rectifyingE_∞ ([SS12]) and E2 ([SS16]) structures respectively.

With the use of then-fold monoidal operads, we have explicitly definedE_nobjects in a diagram category indexed over a smalln-fold monoidal category. In particular we can apply this to the diagram categoryCat^In. In [SS, Theorem 4.11] we showed that the homotopy category of these En diagram categories and the homotopy category of the corresponding E_n structures in Cat are equivalent. A natural

next step is to try and generalize the rectification of E_∞ structures in [SS12] or the rectification of E₂ structures in [SS16] toE_n structures, using the setting of In-categories and In-spaces. As a first step one could start with comparing B-categories andI2-categories to see what can be generalized there.

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[BM03] Clemens Berger and Ieke Moerdijk. Axiomatic homotopy theory for operads.

Comment. Math. Helv., 78(4):805–831, 2003.

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Springer, Berlin, 1970.

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[FSV13] Z. Fiedorowicz, M. Stelzer, and R. M. Vogt. Homotopy colimits of algebras overCat-operads and iterated loop spaces.Adv. Math., 248:1089–1155, 2013.

[FSV16] Zbigniew Fiedorowicz, Manfred Stelzer, and Rainer M. Vogt. Rectification of weak product algebras over an operad in Catand T op and applications.

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[MSS02] Martin Markl, Steve Shnider, and Jim Stasheff. Operads in algebra, topology and physics, volume 96 ofMathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2002.

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[SS] Christian Schlichtkrull and Mirjam Solberg. Higher monoidal injections and diagrammaticEnstructures.

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[SS16] Christian Schlichtkrull and Mirjam Solberg. Braided injections and double loop spaces. Trans. Amer. Math. Soc., 368(10):7305–7338, 2016.

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Included papers

double loop spaces

Christian Schlichtkrull and Mirjam Solberg

Published in: Trans. Amer. Math. Soc. 368 (2016), no. 10, 7305–7338.

Published by the American Mathematical Society.

© 2015 American Mathematical Society.

CHRISTIAN SCHLICHTKRULL AND MIRJAM SOLBERG

Abstract. We consider a framework for representing double loop spaces (and more generallyE2spaces) as commutative monoids. There are analogous commutative rec-tifications of braided monoidal structures and we use this framework to define iterated double deloopings. We also consider commutative rectifications ofE_∞spaces and sym-metric monoidal categories and we relate this to the category of symsym-metric spectra.

1. Introduction

The study of multiplicative structures on spaces has a long history in algebraic topol-ogy. For many spaces of interest the notion of a strictly associative and commutative multiplication is too rigid and must be replaced by the more flexible notion of an E_∞ multiplication encoding higher homotopies between iterated products. This is analogous to the situation for categories where strictly commutative multiplications rarely occur in practice and the more usefulE_∞notion is that of a symmetric monoidal structure. Sim-ilar remarks apply to multiplicative structures on other types of objects. However, for certain kinds of applications it is desirable to be able to replaceE_∞structures by strictly commutative ones, and this can sometimes be achieved by modifying the underlying cate-gory of objects under consideration. An example of this is the introduction of modern cat-egories of spectra (in the sense of stable homotopy theory) [EKMM97, HSS00, MMSS01]

equipped with symmetric monoidal smash products. These categories of spectra have homotopy categories equivalent to the usual stable homotopy category but come with refined multiplicative structures allowing the rectification ofE_∞ring spectra to strictly commutative ring spectra. This has proven useful for the import of ideas and con-structions from commutative algebra into stable homotopy theory. Likewise there are symmetric monoidal refinements of spaces [BCS10, SS12] allowing for analogous rectifi-cations ofE_∞ structures.

Our main objective in this paper is to construct similar commutative rectifications in braided monoidal contexts. In order to provide a setting for this we introduce the categoryBofbraided injections, see Section 2. This is a braided monoidal small category that relates to the category I of finite sets and injections in the same way the braid groups relate to the symmetric groups. We first explain how our rectification works in the setting of small categoriesCat and letBr-Catdenote the category of braided (strict) 2010Mathematics Subject Classification. Primary 18D10, 18D50, 55P48; Secondary 55P43.

Key words and phrases. Braided monoidal categories, double loop spaces, diagram spaces.

First published in: Trans. Amer. Math. Soc. 368 (2016), no. 10, 7305–7338.

c 2015 American Mathematical Society.

1

monoidal small categories. LetCat^B be the diagram category of functors from B to Cat and let us refer to such functors as B-categories. The category Cat^B inherits a braided monoidal convolution product fromB and there is a corresponding category Br-Cat^Bof braided monoidalB-categories. A morphismA→A⁰ inBr-Cat^B is said to be aB-equivalenceif the induced functor of Grothendieck constructionsBR

A→BR A⁰ is a weak equivalence of categories in the usual sense. We write w_B for the class of B-equivalences and wfor the class of morphisms inBr-Cat whose underlying functors are weak equivalences. The following rectification theorem is obtained by combining Proposition 4.12 and Theorem 4.19.

Theorem 1.1.The Grothendieck constructionBR

and the constant embedding∆define an equivalence of the localized categories

BR

:Br-Cat^B[w⁻¹_B]'Br-Cat[w⁻¹] : ∆

and every object in Br-Cat^B is naturally B-equivalent to a strictly commutative B-category monoid.

Thus, working with braided monoidal categories is weakly equivalent to working with braided monoidalB-categories and the latter category has the advantage that we may assume multiplications to be strictly commutative. This implies in particular that every braided monoidal small category is weakly equivalent to one of the form BR

A for a commutativeB-category monoidA.

LetBrbe the categorical operad such that the category ofBr-algebras can be identified withBr-Cat (see Section 5.1 for details). For the analogous rectification in the category of spacesS(which we interpret as the category of simplicial sets) we consider the operad NBr inS obtained by evaluating the nerve ofBr. This is anE2 operad in the sense of being equivalent to the little 2-cubes operad and we may think of the category of algebras NBr-Sas the category ofE2spaces. In order to rectifyE2spaces to strictly commutative monoids we work in the diagram category of B-spaces S^Bequipped with the braided monoidal convolution product inherited from B. There is an analogous category of E2 B-spaces NBr-S^B. After localization with respect to the appropriate classes of B-equivalences wB in NBr-S^B and weak equivalences w in NBr-S, Proposition 5.8 and Theorem 5.9 combine to give the following result.

Theorem 1.2. The homotopy colimit (−)hB and the constant embedding ∆ define an equivalence of the localized categories

(−)hB: NBr-S^B[w⁻¹_B]'NBr-S[w⁻¹] : ∆

and every object inNBr-S^Bis naturallyB-equivalent to a strictly commutativeB-space monoid.

This implies in particular that every double loop space is equivalent to anE₂space of the formA_hBfor a commutativeB-space monoidA. To give an example why this may

be useful, notice that ifAis a commutativeB-space monoid, then the categoryS^B/Aof B-spaces overAinherits the structure of a braided monoidal category. It is less obvious how to define such a structure for the corresponding category of spaces over anE₂space.

The above rectification theorems have corresponding versions for symmetric monoidal categories and E_∞ spaces that we spell out in Section 7. As an application of this we show how to rectify certain E_∞ ring spectra to strictly commutative symmetric ring spectra. However, the braided monoidal setting is somewhat more subtle and is the main focus of this paper.

Our main tool for replacing braided monoidal structures by strictly commutative ones is a refinement of the usual strictification construction used to replace monoidal categories by strictly monoidal ones, see e.g. [JS93, Section 1]. While it is well-known that this construction cannot be used to turn braided monoidal categories into categories with a strictly commutative multiplication, we shall see that it can be reinterpreted so as to take values in commutative B-category monoids instead. This gives rise to the B-category rectification functor Φ introduced in Section 4.14. In order to obtain an analogous rectification on the space level we apply the results of Fiedorowicz-Stelzer-Vogt [FV03, FSV13] that show how to associate braided monoidal categories to E₂ spaces. Our rectification functor Φ then applies to these braided monoidal categories and we can

Our main tool for replacing braided monoidal structures by strictly commutative ones is a refinement of the usual strictification construction used to replace monoidal categories by strictly monoidal ones, see e.g. [JS93, Section 1]. While it is well-known that this construction cannot be used to turn braided monoidal categories into categories with a strictly commutative multiplication, we shall see that it can be reinterpreted so as to take values in commutative B-category monoids instead. This gives rise to the B-category rectification functor Φ introduced in Section 4.14. In order to obtain an analogous rectification on the space level we apply the results of Fiedorowicz-Stelzer-Vogt [FV03, FSV13] that show how to associate braided monoidal categories to E₂ spaces. Our rectification functor Φ then applies to these braided monoidal categories and we can

In document Diagram spaces and multiplicative structures (sider 42-105)